In our study of the stationary complete war of attrition in chapter 4, we saw that all the Nash equilibria are subgame perfect. Similarly, the multiple equilibria just described satisfy the concept of perfect Bayesian equilibrium we introduce in chapter 8.
(Fudenberg&Tirole (1991) p.219)

"stationary" is important and cannot be dropped. Obviously, one player choses "never stop" and the other "always stops" can be Nash equilibrium but which is not subgame perfect.
In chapter 4, they wrote as follows;

All stationary Nash equilibria are subgame perfect. To see this, note that the stationarity of the payoffs implies that all subgame where both players are still active are isomorphic.
(FT p.120-121)